Linear pressure drop coefficient

In case of a Reynolds number of very high value, the linear pressure drop coefficient becomes independent of the Reynolds number. Its value is determined by the relative roughness, which means that turbulence is driven by unevetmess on the wall. 

This alternative formulation enables us to calculate the linear pressure drop coefficient exphdtly and to derive therefiom the value of the mean streamwise velocity. 

Hence, in dealing with steady motion of particulate solids, it is evident that the axial stress or pressure drops exponentially, whereas in the case of liquid flow, it drops linearly with distance. This difference stems, of course, from the fact that frictional forces on the wall are proportional to the absolute local value of normal stress or pressure. In liquids, only the pressure gradient and not the absolute value of the pressure affects the flow. Furthermore, Eq. 4.7.2 indicates that the pushing force increases exponentially with the coefficient of friction and with the geometric, dimensionless group CL/A, which for a tubular conduit becomes 4 L/D. 

Figure 2.63 illustrates the effect of the distortion coefficient (Dc) on the characteristics of a linear and an equal-percentage valve. As the ratio of the minimum to maximum pressure drop increases, the Dc drops and the equal-percentage characteristics of the valve shift toward linear and the linear characteristics shift toward QO. In addition, as the Dc drops, the controllable minimum flow increases, and therefore, the rangeability (the flow range within which the valve characteristic remains as specified) of the valve also drops. 

The pressure drop is given by Darcy s law (eqn.7.1). Optimum flow rates on columns with different particle sizes can be related by using the same reduced velocity (v, eqn.7.4) for each column. Since the diffusion coefficient Dm is a constant, we find for the ratio of the linear velocities on two columns (1 and 2)... 

In addition to these pressure drop models, models to represent spreading of liquid in packed beds because of spatial variation in flow resistance are needed. In a randomly packed bed, the void fraction is not uniform. This implies that some flow channels formed within a packed bed offer less resistance to flow than other channels of equal cross-sectional area. Liquid will tend to move toward channels of lower resistance, leading to higher liquid hold-up in such channels. Thus, even if the initial liquid distribution is uniform, inherent random spatial variation of the bed leads to non-uniform liquid flow. Yin et al. (2000) assumed that the dispersion coefficient for liquid phase volume fraction is linearly proportional to the adverse gradient of... 

Pressure drop and flow rate are usually linearly related - expressed by Darcy s law (Chapter 3.1.4). If unknown from initial tests, the pressure drop is measured for different volume flows, once with column in the plant and once without the column in the plant, using a zero-volume connector. The difference between the two values yields the pressure drop characteristic Apc of the column alone. By plotting Ap, vs. Mjnt the unknown coefficient k0 is readily determined from the slope of the curve by rearranging Eq. 3.7. 

The two-phase flow pattern (in this case, froth) does not change along the length of the pipeline. The liquid-side mass transfer coefficient varies linearly with the superficial gas velocity at a constant liquid flow rate. As indicated later, the pressure drop can be neglected. 

An increase of the linear flow rate of a reaction mixture creates optimal values of the characteristic mixing times of liquid flows, turbulent diffusion coefficients, and dissipation of the specific kinetic energy of turbulence. The upper limit of application of tubular turbulent devices (based on dynamic characteristics of their operation) is evidently the input-output pressure drop in accordance with Ap V, while the lower limit will be determined by the values of the turbulent diffusion coefficient ... 

It is a simple matter to compute coefficients A and B for each peak of interest by linear regression and then find the peaks predicted retention factors at specific temperatures, as shown earlier in this chapter. However, the peaks predicted retention times and not just their retention factors are needed for chromatogram simulation and resolution calculations in the absence of the extensive retention measurements used in the earlier examples. The scenario gets more complex as we combine the effects of temperatme, pressure drop, and column dimensions as required for a more complete column system model. 

Most significantly, the overall pressme loss across the cyclone was found to decrease by 40% at a penetration of about 75% relative to 0% penetration of the vortex finder into the diffuser chamber. Compared to the case where the gas exhausted directly out the vortex finder (the dashed line), the pressure loss decreased by about 44%. These are very significant reductions, especially in light of the simplicity of the diffuser geometry employed in this study. These results support Dehne s earlier comments, only they indicate a far larger pres-sme drop reduction. They are also very similar to those reported by Idelchik (1986) for the pressure loss coefficient for linear (irrotational) flow of a jet exiting a pipe and impacting a flat baffle plate. See Fig. 15.1.17. 

This tendency for preferential gas flows (near the walls) could cause serious problems in systems where the reaction is either strongly exothermic or endothermic, because the uneven temperature distribution may aggravate these bypass phenomena [71,72]. On viewing moving-bed systems within the framework of a fixed coordinate system, it is readily apparent that the linear velocity of the solids is much smaller than that of the gas. It follows that in predicting pressure drop-gas flow relationships, heat and mass transfer coefficients, and the like, we may use the correlations that were previously given in Section 7.3 for fixed-bed systems. 

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